As is well-known the classical action \(S_{\text{c}}\)  for the 1D-HO reads (See e. g. Ref.[1]}):
\(S_{\text{c}}=\frac{m\omega }{2\sin {\omega T}}\left[(x_a^2+x_b^2)\cos {\omega T}-2x_a x_b\right]\) 
